Filterbank optimization with convex objectives and the optimality of principal component forms

This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics. This includes as special cases, many previous results on FB optimization for compression. It also solves problems that have not been considered thus far. FB optimization for coding gain maximization (for compression applications) has been well studied before. The optimum FB has been known to satisfy the principal component property, i.e., it minimizes the mean-square error caused by reconstruction after dropping the P weakest (lowest variance) subbands for any P. We point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of the subband variances produced by the FB. This result has its grounding in majorization and convex function theory and, in particular, explains the optimality of PCFBs for compression. We use the result to show various other optimality properties of PCFBs, especially for noise-suppression applications. Suppose the FB input is a signal corrupted by additive white noise, the desired output is the pure signal, and the subbands of the FB are processed to minimize the output noise. If each subband processor is a zeroth-order Wiener filter for its input, we can show that the expected mean square value of the output noise is a concave function of the subband signal variances. Hence, a PCFB is optimum in the sense of minimizing this mean square error. The above-mentioned concavity of the error and, hence, PCFB optimality, continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of serious practical interest. We prove that certain extensions of this PCFB optimality result to cases where the input noise is colored, and the FB optimization is over a larger class that includes biorthogonal FBs. We also show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs

Discrete multitone modulation with principal component filter banks

Discrete multitone (DMT) modulation is an attractive method for communication over a nonflat channel with possibly colored noise. The uniform discrete Fourier transform (DFT) filter bank and cosine modulated filter bank have in the past been used in this system because of low complexity. We show in this paper that principal component filter banks (PCFB) which are known to be optimal for data compression and denoising applications, are also optimal for a number of criteria in DMT modulation communication. For example, the PCFB of the effective channel noise power spectrum (noise psd weighted by the inverse of the channel gain) is optimal for DMT modulation in the sense of maximizing bit rate for fixed power and error probabilities. We also establish an optimality property of the PCFB when scalar prefilters and postfilters are used around the channel. The difference between the PCFB and a traditional filter bank such as the brickwall filter bank or DFT filter bank is significant for effective power spectra which depart considerably from monotonicity. The twisted pair channel with its bridged taps, next and fext noises, and AM interference, therefore appears to be a good candidate for the application of a PCFB. This is demonstrated with the help of numerical results for the case of the ADSL channel

On the spectral factor ambiguity of FIR energy compaction filter banks

This paper focuses on the design of signal-adapted finite-impulse response (FIR) paraunitary (PU) filter banks optimized for energy compaction (EC). The design of such filter banks has been shown in the literature to consist of the design of an optimal FIR compaction filter followed by an appropriate Karhunen-Loe/spl grave/ve transform (KLT). Despite this elegant construction, EC optimal filter banks have been shown to perform worse than common nonadapted filter banks for coding gain, contrary to intuition. Here, it is shown that this phenomenon is most likely due to the nonuniqueness of the compaction filter in terms of its spectral factors. This nonuniqueness results in a finite set of EC optimal filter banks. By choosing the spectral factor yielding the largest coding gain, it is shown that the resulting filter bank behaves more and more like the infinite-order principal components filter bank (PCFB) in terms of numerous objectives such as coding gain, multiresolution, noise reduction with zeroth-order Wiener filters in the subbands, and power minimization for discrete multitone (DMT)-type nonredundant transmultiplexers

Iterative greedy algorithm for solving the FIR paraunitary approximation problem

In this paper, a method for approximating a multi-input multi-output (MIMO) transfer function by a causal finite-impulse response (FIR) paraunitary (PU) system in a weighted least-squares sense is presented. Using a complete parameterization of FIR PU systems in terms of Householder-like building blocks, an iterative algorithm is proposed that is greedy in the sense that the observed mean-squared error at each iteration is guaranteed to not increase. For certain design problems in which there is a phase-type ambiguity in the desired response, which is formally defined in the paper, a phase feedback modification is proposed in which the phase of the FIR approximant is fed back to the desired response. With this modification in effect, it is shown that the resulting iterative algorithm not only still remains greedy, but also offers a better magnitude-type fit to the desired response. Simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank (PCFB)-like filter banks and the FIR PU interpolation problem. Concerning the PCFB design problem, it is shown that as the McMillan degree of the FIR PU approximant increases, the resulting filter bank behaves more and more like the infinite-order PCFB, consistent with intuition. In particular, this PCFB-like behavior is shown in terms of filter response shape, multiresolution, coding gain, noise reduction with zeroth-order Wiener filtering in the subbands, and power minimization for discrete multitone (DMT)-type transmultiplexers

A state space approach to the design of globally optimal FIR energy compaction filters

We introduce a new approach for the least squared optimization of a weighted FIR filter of arbitrary order N under the constraint that its magnitude squared response be Nyquist(M). Although the new formulation is general enough to cover a wide variety of applications, the focus of the paper is on optimal energy compaction filters. The optimization of such filters has received considerable attention in the past due to the fact that they are the main building blocks in the design of principal component filter banks (PCFBs). The newly proposed method finds the optimum product filter Fopt(z)=Hopt(Z)Hopt (z^-1) corresponding to the compaction filter Hopt (z). By expressing F(z) in the form D(z)+D(z^-1), we show that the compaction problem can be completely parameterized in terms of the state-space realization of the causal function D(z). For a given input power spectrum, the resulting filter Fopt(z) is guaranteed to be a global optimum solution due to the convexity of the new formulation. The new algorithm is universal in the sense that it works for any M, arbitrary filter length N, and any given input power spectrum. Furthermore, additional linear constraints such as wavelets regularity constraints can be incorporated into the design problem. Finally, obtaining Hopt(z) from Fopt(z) does not require an additional spectral factorization step. The minimum-phase spectral factor Hmin(z) can be obtained automatically by relating the state space realization of Dopt(z) to that of H opt(z

Design of FIR paraunitary filter banks for subband coding using a polynomial eigenvalue decomposition

The problem of paraunitary filter bank design for subband coding has received considerable attention in recent years, not least because of the energy preserving property of this class of filter banks. In this paper, we consider the design of signal-adapted, finite impulse response (FIR), paraunitary filter banks using polynomial matrix EVD (PEVD) techniques. Modifications are proposed to an iterative, time-domain PEVD method, known as the sequential best rotation (SBR2) algorithm, which enables its effective application to the problem of FIR orthonormal filter bank design for efficient subband coding. By choosing an optimisation scheme that maximises the coding gain at each stage of the algorithm, it is shown that the resulting filter bank behaves more and more like the infiniteorder principle component filter bank (PCFB). The proposed method is compared to state-of-the-art techniques, namely the iterative greedy algorithm (IGA), the approximate EVD (AEVD), standard SBR2 and a fast algorithm for FIR compaction filter design, called the window method (WM). We demonstrate that for the calculation of the subband coder, the WM approach offers a low-cost alternative at lower coding gains, while at moderate to high complexity, the proposed approach outperforms the benchmarkers. In terms of run-time complexity, AEVD performs well at low orders, while the proposed algorithm offers a better coding gain than the benchmarkers at moderate to high filter order for a number of simulation scenarios

Resource Management in Multicarrier Based Cognitive Radio Systems

The ever-increasing growth of the wireless application and services affirms the importance of the effective usage of the limited radio spectrum. Existing spectrum management policies have led to significant spectrum under-utilization. Recent measurements showed that large range of the spectrum is sparsely used in both temporal and spatial manner. This conflict between the inefficient usage of the spectrum and the continuous evolution in the wireless communication calls upon the development of more flexible management policies. Cognitive radio (CR) with the dynamic spectrum access (DSA) is considered to be a key technology in making the best solution of this conflict by allowing a group of secondary users (SUs) to share the radio spectrum originally allocated to the primary user (PUs). The operation of CR should not negatively alter the performance of the PUs. Therefore, the interference control along with the highly dynamic nature of PUs activities open up new resource allocation problems in CR systems. The resource allocation algorithms should ensure an effective share of the temporarily available frequency bands and deliver the solutions in timely fashion to cope with quick changes in the network. In this dissertation, the resource management problem in multicarrier based CR systems is considered. The dissertation focuses on three main issues: 1) design of efficient resource allocation algorithms to allocate subcarriers and powers between SUs such that no harmful interference is introduced to PUs, 2) compare the spectral efficiency of using different multicarrier schemes in the CR physical layer, specifically, orthogonal frequency division multiplexing (OFDM) and filter bank multicarrier (FBMC) schemes, 3) investigate the impact of the different constraints values on the overall performance of the CR system. Three different scenarios are considered in this dissertation, namely downlink transmission, uplink transmission, and relayed transmission. For every scenario, the optimal solution is examined and efficient sub-optimal algorithms are proposed to reduce the computational burden of obtaining the optimal solution. The suboptimal algorithms are developed by separate the subcarrier and power allocation into two steps in downlink and uplink scenarios. In the relayed scenario, dual decomposition technique is used to obtain an asymptotically optimal solution, and a joint heuristic algorithm is proposed to find the suboptimal solution. Numerical simulations show that the proposed suboptimal algorithms achieve a near optimal performance and perform better than the existing algorithms designed for cognitive and non-cognitive systems. Eventually, the ability of FBMC to overcome the OFDM drawbacks and achieve more spectral efficiency is verified which recommends the consideration of FBMC in the future CR systems.El crecimiento continuo de las aplicaciones y servicios en sistemas inal´ambricos, indica la importancia y necesidad de una utilizaci´on eficaz del espectro radio. Las pol´ıticas actuales de gesti´on del espectro han conducido a una infrautilizaci´on del propio espectro radioel´ectrico. Recientes mediciones en diferentes entornos han mostrado que gran parte del espectro queda poco utilizado en sus ambas vertientes, la temporal, y la espacial. El permanente conflicto entre el uso ineficiente del espectro y la evoluci´on continua de los sistemas de comunicaci´on inal´ambrica, hace que sea urgente y necesario el desarrollo de esquemas de gesti´on del espectro m´as flexibles. Se considera el acceso din´amico (DSA) al espectro en los sistemas cognitivos como una tecnolog´ıa clave para resolver este conflicto al permitir que un grupo de usuarios secundarios (SUs) puedan compartir y acceder al espectro asignado inicialmente a uno o varios usuarios primarios (PUs). Las operaciones de comunicaci´on llevadas a cabo por los sistemas radio cognitivos no deben en ning´un caso alterar (interferir) los sistemas primarios. Por tanto, el control de la interferencia junto al gran dinamismo de los sistemas primarios implica nuevos retos en el control y asignaci´on de los recursos radio en los sistemas de comunicaci´on CR. Los algoritmos de gesti´on y asignaci´on de recursos (Radio Resource Management-RRM) deben garantizar una participaci´on efectiva de las bandas con frecuencias disponibles temporalmente, y ofrecer en cada momento oportunas soluciones para hacer frente a los distintos cambios r´apidos que influyen en la misma red. En esta tesis doctoral, se analiza el problema de la gesti´on de los recursos radio en sistemas multiportadoras CR, proponiendo varias soluciones para su uso eficaz y coexistencia con los PUs. La tesis en s´ı, se centra en tres l´ıneas principales: 1) el dise˜no de algoritmos eficientes de gesti´on de recursos para la asignaci´on de sub-portadoras y distribuci´on de la potencia en sistemas segundarios, evitando asi cualquier interferencia que pueda ser perjudicial para el funcionamiento normal de los usuarios de la red primaria, 2) analizar y comparar la eficiencia espectral alcanzada a la hora de utilizar diferentes esquema de transmisi´on multiportadora en la capa f´ısica del sistema CR, espec´ıficamente en sistemas basados en OFDM y los basados en banco de filtros multiportadoras (Filter bank Multicarrier-FBMC), 3) investigar el impacto de las diferentes limitaciones en el rendimiento total del sistema de CR. Los escenarios considerados en esta tesis son tres, es decir; modo de transmisi´on descendente (downlink), modo de transmisi´on ascendente (uplink), y el modo de transmisi´on ”Relay”. En cada escenario, la soluci´on ´optima es examinada y comparada con algoritmos sub- ´optimos que tienen como objetivo principal reducir la carga computacional. Los algoritmos sub-´optimos son llevados a cabo en dos fases mediante la separaci´on del propio proceso de distribuci´on de subportadoras y la asignaci´on de la potencia en los modos de comunicaci´on descendente (downlink), y ascendente (uplink). Para los entornos de tipo ”Relay”, se ha utilizado la t´ecnica de doble descomposici´on (dual decomposition) para obtener una soluci´on asint´oticamente ´optima. Adem´as, se ha desarrollado un algoritmo heur´ıstico para poder obtener la soluci´on ´optima con un reducido coste computacional. Los resultados obtenidos mediante simulaciones num´ericas muestran que los algoritmos sub-´optimos desarrollados logran acercarse a la soluci´on ´optima en cada uno de los entornos analizados, logrando as´ı un mayor rendimiento que los ya existentes y utilizados tanto en entornos cognitivos como no-cognitivos. Se puede comprobar en varios resultados obtenidos en la tesis la superioridad del esquema multiportadora FBMC sobre los sistemas basados en OFDM para los entornos cognitivos, causando una menor interferencia que el OFDM en los sistemas primarios, y logrando una mayor eficiencia espectral. Finalmente, en base a lo analizado en esta tesis, podemos recomendar al esquema multiportadora FBMC como una id´onea y potente forma de comunicaci´on para las futuras redes cognitivas

Adaptive sparse coding and dictionary selection

Grant no. D000246/1.The sparse coding is approximation/representation of signals with the minimum number of coefficients using an overcomplete set of elementary functions. This kind of approximations/ representations has found numerous applications in source separation, denoising, coding and compressed sensing. The adaptation of the sparse approximation framework to the coding problem of signals is investigated in this thesis. Open problems are the selection of appropriate models and their orders, coefficient quantization and sparse approximation method. Some of these questions are addressed in this thesis and novel methods developed. Because almost all recent communication and storage systems are digital, an easy method to compute quantized sparse approximations is introduced in the first part. The model selection problem is investigated next. The linear model can be adapted to better fit a given signal class. It can also be designed based on some a priori information about the model. Two novel dictionary selection methods are separately presented in the second part of the thesis. The proposed model adaption algorithm, called Dictionary Learning with the Majorization Method (DLMM), is much more general than current methods. This generality allowes it to be used with different constraints on the model. Particularly, two important cases have been considered in this thesis for the first time, Parsimonious Dictionary Learning (PDL) and Compressible Dictionary Learning (CDL). When the generative model order is not given, PDL not only adapts the dictionary to the given class of signals, but also reduces the model order redundancies. When a fast dictionary is needed, the CDL framework helps us to find a dictionary which is adapted to the given signal class without increasing the computation cost so much. Sometimes a priori information about the linear generative model is given in format of a parametric function. Parametric Dictionary Design (PDD) generates a suitable dictionary for sparse coding using the parametric function. Basically PDD finds a parametric dictionary with a minimal dictionary coherence, which has been shown to be suitable for sparse approximation and exact sparse recovery. Theoretical analyzes are accompanied by experiments to validate the analyzes. This research was primarily used for audio applications, as audio can be shown to have sparse structures. Therefore, most of the experiments are done using audio signals